3.610 \(\int \frac{(f+g x) (a+b \log (c (d+e x^2)^p))}{(h x)^{9/2}} \, dx\)

Optimal. Leaf size=641 \[ -\frac{2 f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{7 h (h x)^{7/2}}-\frac{2 g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h^2 (h x)^{5/2}}+\frac{\sqrt{2} b e^{7/4} f p \log \left (-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}+\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x\right )}{7 d^{7/4} h^{9/2}}-\frac{\sqrt{2} b e^{7/4} f p \log \left (\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}+\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x\right )}{7 d^{7/4} h^{9/2}}+\frac{2 \sqrt{2} b e^{7/4} f p \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{7 d^{7/4} h^{9/2}}-\frac{2 \sqrt{2} b e^{7/4} f p \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}+1\right )}{7 d^{7/4} h^{9/2}}-\frac{\sqrt{2} b e^{5/4} g p \log \left (-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}+\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x\right )}{5 d^{5/4} h^{9/2}}+\frac{\sqrt{2} b e^{5/4} g p \log \left (\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}+\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x\right )}{5 d^{5/4} h^{9/2}}+\frac{2 \sqrt{2} b e^{5/4} g p \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{5 d^{5/4} h^{9/2}}-\frac{2 \sqrt{2} b e^{5/4} g p \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}+1\right )}{5 d^{5/4} h^{9/2}}-\frac{8 b e f p}{21 d h^3 (h x)^{3/2}}-\frac{8 b e g p}{5 d h^4 \sqrt{h x}} \]

[Out]

(-8*b*e*f*p)/(21*d*h^3*(h*x)^(3/2)) - (8*b*e*g*p)/(5*d*h^4*Sqrt[h*x]) + (2*Sqrt[2]*b*e^(7/4)*f*p*ArcTan[1 - (S
qrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(7*d^(7/4)*h^(9/2)) + (2*Sqrt[2]*b*e^(5/4)*g*p*ArcTan[1 - (Sqrt[
2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(5*d^(5/4)*h^(9/2)) - (2*Sqrt[2]*b*e^(7/4)*f*p*ArcTan[1 + (Sqrt[2]*e
^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(7*d^(7/4)*h^(9/2)) - (2*Sqrt[2]*b*e^(5/4)*g*p*ArcTan[1 + (Sqrt[2]*e^(1/
4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(5*d^(5/4)*h^(9/2)) - (2*f*(a + b*Log[c*(d + e*x^2)^p]))/(7*h*(h*x)^(7/2)) -
(2*g*(a + b*Log[c*(d + e*x^2)^p]))/(5*h^2*(h*x)^(5/2)) + (Sqrt[2]*b*e^(7/4)*f*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]
*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(7*d^(7/4)*h^(9/2)) - (Sqrt[2]*b*e^(5/4)*g*p*Log[Sqrt[d]*Sqrt
[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(5*d^(5/4)*h^(9/2)) - (Sqrt[2]*b*e^(7/4)*f*p*Log
[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(7*d^(7/4)*h^(9/2)) + (Sqrt[2]*b*e^
(5/4)*g*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(5*d^(5/4)*h^(9/2))

________________________________________________________________________________________

Rubi [A]  time = 0.826927, antiderivative size = 641, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 11, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.379, Rules used = {2467, 2476, 2455, 325, 211, 1165, 628, 1162, 617, 204, 297} \[ -\frac{2 f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{7 h (h x)^{7/2}}-\frac{2 g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h^2 (h x)^{5/2}}+\frac{\sqrt{2} b e^{7/4} f p \log \left (-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}+\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x\right )}{7 d^{7/4} h^{9/2}}-\frac{\sqrt{2} b e^{7/4} f p \log \left (\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}+\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x\right )}{7 d^{7/4} h^{9/2}}+\frac{2 \sqrt{2} b e^{7/4} f p \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{7 d^{7/4} h^{9/2}}-\frac{2 \sqrt{2} b e^{7/4} f p \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}+1\right )}{7 d^{7/4} h^{9/2}}-\frac{\sqrt{2} b e^{5/4} g p \log \left (-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}+\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x\right )}{5 d^{5/4} h^{9/2}}+\frac{\sqrt{2} b e^{5/4} g p \log \left (\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}+\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x\right )}{5 d^{5/4} h^{9/2}}+\frac{2 \sqrt{2} b e^{5/4} g p \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{5 d^{5/4} h^{9/2}}-\frac{2 \sqrt{2} b e^{5/4} g p \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}+1\right )}{5 d^{5/4} h^{9/2}}-\frac{8 b e f p}{21 d h^3 (h x)^{3/2}}-\frac{8 b e g p}{5 d h^4 \sqrt{h x}} \]

Antiderivative was successfully verified.

[In]

Int[((f + g*x)*(a + b*Log[c*(d + e*x^2)^p]))/(h*x)^(9/2),x]

[Out]

(-8*b*e*f*p)/(21*d*h^3*(h*x)^(3/2)) - (8*b*e*g*p)/(5*d*h^4*Sqrt[h*x]) + (2*Sqrt[2]*b*e^(7/4)*f*p*ArcTan[1 - (S
qrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(7*d^(7/4)*h^(9/2)) + (2*Sqrt[2]*b*e^(5/4)*g*p*ArcTan[1 - (Sqrt[
2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(5*d^(5/4)*h^(9/2)) - (2*Sqrt[2]*b*e^(7/4)*f*p*ArcTan[1 + (Sqrt[2]*e
^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(7*d^(7/4)*h^(9/2)) - (2*Sqrt[2]*b*e^(5/4)*g*p*ArcTan[1 + (Sqrt[2]*e^(1/
4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(5*d^(5/4)*h^(9/2)) - (2*f*(a + b*Log[c*(d + e*x^2)^p]))/(7*h*(h*x)^(7/2)) -
(2*g*(a + b*Log[c*(d + e*x^2)^p]))/(5*h^2*(h*x)^(5/2)) + (Sqrt[2]*b*e^(7/4)*f*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]
*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(7*d^(7/4)*h^(9/2)) - (Sqrt[2]*b*e^(5/4)*g*p*Log[Sqrt[d]*Sqrt
[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(5*d^(5/4)*h^(9/2)) - (Sqrt[2]*b*e^(7/4)*f*p*Log
[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(7*d^(7/4)*h^(9/2)) + (Sqrt[2]*b*e^
(5/4)*g*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(5*d^(5/4)*h^(9/2))

Rule 2467

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))^(p_.)]*(b_.))^(q_.)*((h_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(r
_.), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/h, Subst[Int[x^(k*(m + 1) - 1)*(f + (g*x^k)/h)^r*(a + b*Lo
g[c*(d + (e*x^(k*n))/h^n)^p])^q, x], x, (h*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, f, g, h, p, r}, x] && Fract
ionQ[m] && IntegerQ[n] && IntegerQ[r]

Rule 2476

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x^s)^r, x], x] /; FreeQ[{a, b, c,
d, e, f, g, m, n, p, q, r, s}, x] && IGtQ[q, 0] && IntegerQ[m] && IntegerQ[r] && IntegerQ[s]

Rule 2455

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[((f*x)^(m
+ 1)*(a + b*Log[c*(d + e*x^n)^p]))/(f*(m + 1)), x] - Dist[(b*e*n*p)/(f*(m + 1)), Int[(x^(n - 1)*(f*x)^(m + 1))
/(d + e*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 211

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 297

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rubi steps

\begin{align*} \int \frac{(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{9/2}} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{\left (f+\frac{g x^2}{h}\right ) \left (a+b \log \left (c \left (d+\frac{e x^4}{h^2}\right )^p\right )\right )}{x^8} \, dx,x,\sqrt{h x}\right )}{h}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (\frac{f \left (a+b \log \left (c \left (d+\frac{e x^4}{h^2}\right )^p\right )\right )}{x^8}+\frac{g \left (a+b \log \left (c \left (d+\frac{e x^4}{h^2}\right )^p\right )\right )}{h x^6}\right ) \, dx,x,\sqrt{h x}\right )}{h}\\ &=\frac{(2 g) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e x^4}{h^2}\right )^p\right )}{x^6} \, dx,x,\sqrt{h x}\right )}{h^2}+\frac{(2 f) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e x^4}{h^2}\right )^p\right )}{x^8} \, dx,x,\sqrt{h x}\right )}{h}\\ &=-\frac{2 f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{7 h (h x)^{7/2}}-\frac{2 g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h^2 (h x)^{5/2}}+\frac{(8 b e g p) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (d+\frac{e x^4}{h^2}\right )} \, dx,x,\sqrt{h x}\right )}{5 h^4}+\frac{(8 b e f p) \operatorname{Subst}\left (\int \frac{1}{x^4 \left (d+\frac{e x^4}{h^2}\right )} \, dx,x,\sqrt{h x}\right )}{7 h^3}\\ &=-\frac{8 b e f p}{21 d h^3 (h x)^{3/2}}-\frac{8 b e g p}{5 d h^4 \sqrt{h x}}-\frac{2 f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{7 h (h x)^{7/2}}-\frac{2 g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h^2 (h x)^{5/2}}-\frac{\left (8 b e^2 g p\right ) \operatorname{Subst}\left (\int \frac{x^2}{d+\frac{e x^4}{h^2}} \, dx,x,\sqrt{h x}\right )}{5 d h^6}-\frac{\left (8 b e^2 f p\right ) \operatorname{Subst}\left (\int \frac{1}{d+\frac{e x^4}{h^2}} \, dx,x,\sqrt{h x}\right )}{7 d h^5}\\ &=-\frac{8 b e f p}{21 d h^3 (h x)^{3/2}}-\frac{8 b e g p}{5 d h^4 \sqrt{h x}}-\frac{2 f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{7 h (h x)^{7/2}}-\frac{2 g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h^2 (h x)^{5/2}}-\frac{\left (4 b e^2 f p\right ) \operatorname{Subst}\left (\int \frac{\sqrt{d} h-\sqrt{e} x^2}{d+\frac{e x^4}{h^2}} \, dx,x,\sqrt{h x}\right )}{7 d^{3/2} h^6}-\frac{\left (4 b e^2 f p\right ) \operatorname{Subst}\left (\int \frac{\sqrt{d} h+\sqrt{e} x^2}{d+\frac{e x^4}{h^2}} \, dx,x,\sqrt{h x}\right )}{7 d^{3/2} h^6}+\frac{\left (4 b e^{3/2} g p\right ) \operatorname{Subst}\left (\int \frac{\sqrt{d} h-\sqrt{e} x^2}{d+\frac{e x^4}{h^2}} \, dx,x,\sqrt{h x}\right )}{5 d h^6}-\frac{\left (4 b e^{3/2} g p\right ) \operatorname{Subst}\left (\int \frac{\sqrt{d} h+\sqrt{e} x^2}{d+\frac{e x^4}{h^2}} \, dx,x,\sqrt{h x}\right )}{5 d h^6}\\ &=-\frac{8 b e f p}{21 d h^3 (h x)^{3/2}}-\frac{8 b e g p}{5 d h^4 \sqrt{h x}}-\frac{2 f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{7 h (h x)^{7/2}}-\frac{2 g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h^2 (h x)^{5/2}}+\frac{\left (\sqrt{2} b e^{7/4} f p\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{d} \sqrt{h}}{\sqrt [4]{e}}+2 x}{-\frac{\sqrt{d} h}{\sqrt{e}}-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{h} x}{\sqrt [4]{e}}-x^2} \, dx,x,\sqrt{h x}\right )}{7 d^{7/4} h^{9/2}}+\frac{\left (\sqrt{2} b e^{7/4} f p\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{d} \sqrt{h}}{\sqrt [4]{e}}-2 x}{-\frac{\sqrt{d} h}{\sqrt{e}}+\frac{\sqrt{2} \sqrt [4]{d} \sqrt{h} x}{\sqrt [4]{e}}-x^2} \, dx,x,\sqrt{h x}\right )}{7 d^{7/4} h^{9/2}}-\frac{\left (\sqrt{2} b e^{5/4} g p\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{d} \sqrt{h}}{\sqrt [4]{e}}+2 x}{-\frac{\sqrt{d} h}{\sqrt{e}}-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{h} x}{\sqrt [4]{e}}-x^2} \, dx,x,\sqrt{h x}\right )}{5 d^{5/4} h^{9/2}}-\frac{\left (\sqrt{2} b e^{5/4} g p\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{d} \sqrt{h}}{\sqrt [4]{e}}-2 x}{-\frac{\sqrt{d} h}{\sqrt{e}}+\frac{\sqrt{2} \sqrt [4]{d} \sqrt{h} x}{\sqrt [4]{e}}-x^2} \, dx,x,\sqrt{h x}\right )}{5 d^{5/4} h^{9/2}}-\frac{\left (2 b e^{3/2} f p\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{d} h}{\sqrt{e}}-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{h} x}{\sqrt [4]{e}}+x^2} \, dx,x,\sqrt{h x}\right )}{7 d^{3/2} h^4}-\frac{\left (2 b e^{3/2} f p\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{d} h}{\sqrt{e}}+\frac{\sqrt{2} \sqrt [4]{d} \sqrt{h} x}{\sqrt [4]{e}}+x^2} \, dx,x,\sqrt{h x}\right )}{7 d^{3/2} h^4}-\frac{(2 b e g p) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{d} h}{\sqrt{e}}-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{h} x}{\sqrt [4]{e}}+x^2} \, dx,x,\sqrt{h x}\right )}{5 d h^4}-\frac{(2 b e g p) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{d} h}{\sqrt{e}}+\frac{\sqrt{2} \sqrt [4]{d} \sqrt{h} x}{\sqrt [4]{e}}+x^2} \, dx,x,\sqrt{h x}\right )}{5 d h^4}\\ &=-\frac{8 b e f p}{21 d h^3 (h x)^{3/2}}-\frac{8 b e g p}{5 d h^4 \sqrt{h x}}-\frac{2 f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{7 h (h x)^{7/2}}-\frac{2 g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h^2 (h x)^{5/2}}+\frac{\sqrt{2} b e^{7/4} f p \log \left (\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}\right )}{7 d^{7/4} h^{9/2}}-\frac{\sqrt{2} b e^{5/4} g p \log \left (\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}\right )}{5 d^{5/4} h^{9/2}}-\frac{\sqrt{2} b e^{7/4} f p \log \left (\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x+\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}\right )}{7 d^{7/4} h^{9/2}}+\frac{\sqrt{2} b e^{5/4} g p \log \left (\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x+\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}\right )}{5 d^{5/4} h^{9/2}}-\frac{\left (2 \sqrt{2} b e^{7/4} f p\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{7 d^{7/4} h^{9/2}}+\frac{\left (2 \sqrt{2} b e^{7/4} f p\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{7 d^{7/4} h^{9/2}}-\frac{\left (2 \sqrt{2} b e^{5/4} g p\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{5 d^{5/4} h^{9/2}}+\frac{\left (2 \sqrt{2} b e^{5/4} g p\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{5 d^{5/4} h^{9/2}}\\ &=-\frac{8 b e f p}{21 d h^3 (h x)^{3/2}}-\frac{8 b e g p}{5 d h^4 \sqrt{h x}}+\frac{2 \sqrt{2} b e^{7/4} f p \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{7 d^{7/4} h^{9/2}}+\frac{2 \sqrt{2} b e^{5/4} g p \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{5 d^{5/4} h^{9/2}}-\frac{2 \sqrt{2} b e^{7/4} f p \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{7 d^{7/4} h^{9/2}}-\frac{2 \sqrt{2} b e^{5/4} g p \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{5 d^{5/4} h^{9/2}}-\frac{2 f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{7 h (h x)^{7/2}}-\frac{2 g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h^2 (h x)^{5/2}}+\frac{\sqrt{2} b e^{7/4} f p \log \left (\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}\right )}{7 d^{7/4} h^{9/2}}-\frac{\sqrt{2} b e^{5/4} g p \log \left (\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}\right )}{5 d^{5/4} h^{9/2}}-\frac{\sqrt{2} b e^{7/4} f p \log \left (\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x+\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}\right )}{7 d^{7/4} h^{9/2}}+\frac{\sqrt{2} b e^{5/4} g p \log \left (\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x+\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}\right )}{5 d^{5/4} h^{9/2}}\\ \end{align*}

Mathematica [C]  time = 0.0687117, size = 100, normalized size = 0.16 \[ -\frac{2 \sqrt{h x} \left (3 d (5 f+7 g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )+20 b e f p x^2 \, _2F_1\left (-\frac{3}{4},1;\frac{1}{4};-\frac{e x^2}{d}\right )+84 b e g p x^3 \, _2F_1\left (-\frac{1}{4},1;\frac{3}{4};-\frac{e x^2}{d}\right )\right )}{105 d h^5 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[((f + g*x)*(a + b*Log[c*(d + e*x^2)^p]))/(h*x)^(9/2),x]

[Out]

(-2*Sqrt[h*x]*(20*b*e*f*p*x^2*Hypergeometric2F1[-3/4, 1, 1/4, -((e*x^2)/d)] + 84*b*e*g*p*x^3*Hypergeometric2F1
[-1/4, 1, 3/4, -((e*x^2)/d)] + 3*d*(5*f + 7*g*x)*(a + b*Log[c*(d + e*x^2)^p])))/(105*d*h^5*x^4)

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Maple [F]  time = 1.201, size = 0, normalized size = 0. \begin{align*} \int{ \left ( gx+f \right ) \left ( a+b\ln \left ( c \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) \left ( hx \right ) ^{-{\frac{9}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)*(a+b*ln(c*(e*x^2+d)^p))/(h*x)^(9/2),x)

[Out]

int((g*x+f)*(a+b*ln(c*(e*x^2+d)^p))/(h*x)^(9/2),x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(a+b*log(c*(e*x^2+d)^p))/(h*x)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.12623, size = 2996, normalized size = 4.67 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(a+b*log(c*(e*x^2+d)^p))/(h*x)^(9/2),x, algorithm="fricas")

[Out]

2/105*(3*d*h^5*x^4*sqrt(-(d^3*h^9*sqrt(-(625*b^4*e^7*f^4 - 2450*b^4*d*e^6*f^2*g^2 + 2401*b^4*d^2*e^5*g^4)*p^4/
(d^7*h^18)) + 70*b^2*e^3*f*g*p^2)/(d^3*h^9))*log(-32*(625*b^3*e^6*f^4 - 2401*b^3*d^2*e^4*g^4)*sqrt(h*x)*p^3 +
32*(7*d^6*g*h^14*sqrt(-(625*b^4*e^7*f^4 - 2450*b^4*d*e^6*f^2*g^2 + 2401*b^4*d^2*e^5*g^4)*p^4/(d^7*h^18)) + 5*(
25*b^2*d^2*e^4*f^3 - 49*b^2*d^3*e^3*f*g^2)*h^5*p^2)*sqrt(-(d^3*h^9*sqrt(-(625*b^4*e^7*f^4 - 2450*b^4*d*e^6*f^2
*g^2 + 2401*b^4*d^2*e^5*g^4)*p^4/(d^7*h^18)) + 70*b^2*e^3*f*g*p^2)/(d^3*h^9))) - 3*d*h^5*x^4*sqrt(-(d^3*h^9*sq
rt(-(625*b^4*e^7*f^4 - 2450*b^4*d*e^6*f^2*g^2 + 2401*b^4*d^2*e^5*g^4)*p^4/(d^7*h^18)) + 70*b^2*e^3*f*g*p^2)/(d
^3*h^9))*log(-32*(625*b^3*e^6*f^4 - 2401*b^3*d^2*e^4*g^4)*sqrt(h*x)*p^3 - 32*(7*d^6*g*h^14*sqrt(-(625*b^4*e^7*
f^4 - 2450*b^4*d*e^6*f^2*g^2 + 2401*b^4*d^2*e^5*g^4)*p^4/(d^7*h^18)) + 5*(25*b^2*d^2*e^4*f^3 - 49*b^2*d^3*e^3*
f*g^2)*h^5*p^2)*sqrt(-(d^3*h^9*sqrt(-(625*b^4*e^7*f^4 - 2450*b^4*d*e^6*f^2*g^2 + 2401*b^4*d^2*e^5*g^4)*p^4/(d^
7*h^18)) + 70*b^2*e^3*f*g*p^2)/(d^3*h^9))) - 3*d*h^5*x^4*sqrt((d^3*h^9*sqrt(-(625*b^4*e^7*f^4 - 2450*b^4*d*e^6
*f^2*g^2 + 2401*b^4*d^2*e^5*g^4)*p^4/(d^7*h^18)) - 70*b^2*e^3*f*g*p^2)/(d^3*h^9))*log(-32*(625*b^3*e^6*f^4 - 2
401*b^3*d^2*e^4*g^4)*sqrt(h*x)*p^3 + 32*(7*d^6*g*h^14*sqrt(-(625*b^4*e^7*f^4 - 2450*b^4*d*e^6*f^2*g^2 + 2401*b
^4*d^2*e^5*g^4)*p^4/(d^7*h^18)) - 5*(25*b^2*d^2*e^4*f^3 - 49*b^2*d^3*e^3*f*g^2)*h^5*p^2)*sqrt((d^3*h^9*sqrt(-(
625*b^4*e^7*f^4 - 2450*b^4*d*e^6*f^2*g^2 + 2401*b^4*d^2*e^5*g^4)*p^4/(d^7*h^18)) - 70*b^2*e^3*f*g*p^2)/(d^3*h^
9))) + 3*d*h^5*x^4*sqrt((d^3*h^9*sqrt(-(625*b^4*e^7*f^4 - 2450*b^4*d*e^6*f^2*g^2 + 2401*b^4*d^2*e^5*g^4)*p^4/(
d^7*h^18)) - 70*b^2*e^3*f*g*p^2)/(d^3*h^9))*log(-32*(625*b^3*e^6*f^4 - 2401*b^3*d^2*e^4*g^4)*sqrt(h*x)*p^3 - 3
2*(7*d^6*g*h^14*sqrt(-(625*b^4*e^7*f^4 - 2450*b^4*d*e^6*f^2*g^2 + 2401*b^4*d^2*e^5*g^4)*p^4/(d^7*h^18)) - 5*(2
5*b^2*d^2*e^4*f^3 - 49*b^2*d^3*e^3*f*g^2)*h^5*p^2)*sqrt((d^3*h^9*sqrt(-(625*b^4*e^7*f^4 - 2450*b^4*d*e^6*f^2*g
^2 + 2401*b^4*d^2*e^5*g^4)*p^4/(d^7*h^18)) - 70*b^2*e^3*f*g*p^2)/(d^3*h^9))) - (84*b*e*g*p*x^3 + 20*b*e*f*p*x^
2 + 21*a*d*g*x + 15*a*d*f + 3*(7*b*d*g*p*x + 5*b*d*f*p)*log(e*x^2 + d) + 3*(7*b*d*g*x + 5*b*d*f)*log(c))*sqrt(
h*x))/(d*h^5*x^4)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(a+b*ln(c*(e*x**2+d)**p))/(h*x)**(9/2),x)

[Out]

Timed out

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Giac [A]  time = 1.61487, size = 659, normalized size = 1.03 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(a+b*log(c*(e*x^2+d)^p))/(h*x)^(9/2),x, algorithm="giac")

[Out]

-1/35*(2*(5*sqrt(2)*(d*h^2)^(1/4)*b*f*h*p*e^(11/4) + 7*sqrt(2)*(d*h^2)^(3/4)*b*g*p*e^(9/4))*arctan(1/2*sqrt(2)
*(sqrt(2)*(d*h^2)^(1/4)*e^(-1/4) + 2*sqrt(h*x))*e^(1/4)/(d*h^2)^(1/4))*e^(-1)/(d^2*h^2) + 2*(5*sqrt(2)*(d*h^2)
^(1/4)*b*f*h*p*e^(11/4) + 7*sqrt(2)*(d*h^2)^(3/4)*b*g*p*e^(9/4))*arctan(-1/2*sqrt(2)*(sqrt(2)*(d*h^2)^(1/4)*e^
(-1/4) - 2*sqrt(h*x))*e^(1/4)/(d*h^2)^(1/4))*e^(-1)/(d^2*h^2) + (5*sqrt(2)*(d*h^2)^(1/4)*b*f*h*p*e^(11/4) - 7*
sqrt(2)*(d*h^2)^(3/4)*b*g*p*e^(9/4))*e^(-1)*log(sqrt(2)*(d*h^2)^(1/4)*sqrt(h*x)*e^(-1/4) + h*x + sqrt(d*h^2)*e
^(-1/2))/(d^2*h^2) - (5*sqrt(2)*(d*h^2)^(1/4)*b*f*h*p*e^(11/4) - 7*sqrt(2)*(d*h^2)^(3/4)*b*g*p*e^(9/4))*e^(-1)
*log(-sqrt(2)*(d*h^2)^(1/4)*sqrt(h*x)*e^(-1/4) + h*x + sqrt(d*h^2)*e^(-1/2))/(d^2*h^2))/h^4 - 2/105*(84*b*g*h^
4*p*x^3*e + 20*b*f*h^4*p*x^2*e + 21*b*d*g*h^4*p*x*log(h^2*x^2*e + d*h^2) - 21*b*d*g*h^4*p*x*log(h^2) + 15*b*d*
f*h^4*p*log(h^2*x^2*e + d*h^2) - 15*b*d*f*h^4*p*log(h^2) + 21*b*d*g*h^4*x*log(c) + 21*a*d*g*h^4*x + 15*b*d*f*h
^4*log(c) + 15*a*d*f*h^4)/(sqrt(h*x)*d*h^8*x^3)